 Friends, in the last lecture we looked at Weiss's Prater criterion for deciding from the experimental information whether the reaction is internal diffusion controlled or not. So, in this lecture we look at what is the reason why Weiss's Prater scheme works and what are the limitations of the scheme and what is the correction for the same. So the Weiss's Prater criterion is basically uses a parameter called CWP which is equal to the internal effectiveness factor multiplied by the Thiele modulus square of phi square and that is equal to the R observed, the observed reaction rate multiplied by the density of the catalyst multiplied by the square of the length scale of the pellet that is being used divided by the diffusivity into the corresponding concentration of the species at the surface of the catalyst. So, if this is less than 1 then this suggests that there is no internal diffusional limitations. The reaction is not limited by the internal diffusion. So, now the question is what is the validity of this criteria, when does it work, does it work for all reactions and all catalytic reactions. So, in order to understand this let us look at why the Weiss's Prater criteria works, why is it being, why is it the correct criteria in order to estimate whether the diffusion limitations are present. So, in order to establish that let us look at the classical plot of Thiele modulus versus the internal effectiveness factor eta. So, now if it is a first order reaction, if it is a zero order reaction then this is for the zero order reaction, zeroth order reaction. Now, if it is a first order reaction the curve looks like this and if it is a second order reaction the curve looks like this and so on. So, this is the first order reaction, this is for the second order reaction and And you can now look at other nth order reaction. So from this graph one can decipher that there is no internal diffusion limitations if when the internal effectiveness factor is approximately equal to 1. So the effectiveness factor here starts at 1. So it is approximately when it is approximately equal to 1 then it means that there is no internal diffusional limitations. Now from the graph one can easily decipher that when eta is approximately equal to 1 or slightly less than 1 then the Thiele model is phi is less than 1. So therefore clearly eta times phi square should be less than 1 so which is the which is the vice-prater parameter. So the vice-prater parameter this is basically the vice-prater criterion. So therefore as long as the eta phi relationship behaves the way as it is depicted in this picture the vice-prater criterion would usually work. And in fact the eta versus phi curve it looks like this only for typically for an nth order reaction any n for nth order reaction. If it is not an nth order reaction for example if there is adsorption of a species or product inhibition or if it is a non-isothermal then the eta versus phi can be different from what is depicted in this picture here. Does not mean that it will not follow this picture but if it approximately follows an nth order reaction then this is the kind of profile that one would get. Now if suppose if it is not a simple nth order reaction then it is possible that the eta versus phi graph will not look like this and therefore this condition of eta phi square less than 1 is not always valid according to the definition that is given by vice-prater criterion that is the vice-prater parameter the vice-prater parameter Cwp equal to minus rA observed reaction rate into the density of the catalyst into square of the length scale divided by the diffusivity into CAS. So this being less than 1 is not always valid if the Thiele modulus and the internal effectiveness graph does not look like the way we just depicted. Let us look at what happens if there is a reaction which does not necessarily follow such a effectiveness factor Thiele modulus relationship. One particular example is the reaction of carbon when it reacts with the carbon dioxide it leaves out 2 moles of carbon monoxide. So suppose I represent this as B plus A giving 2D so that is the reaction. So if I say B species B is carbon species A is CO2 gas and species D is carbon monoxide. So at 1000 Kelvin which is where the reaction is conducted Austin and Walker in 1963 measured the rate of reaction rate and other parameters. So the reaction rate the observed reaction rate multiplied by the density of the catalyst was found to be 4.67 into 10 power minus 9 moles per centimeter cube second. So that was the reaction rate that was observed and the diffusivity of species A effective diffusivity of species A is about 1.1 centimeter square per second and the effective radius radius of the particle pellet that was used is 0.7 centimeters. So this information is basically measured by the group of Austin and Walker in 1963. So the data was measured by these two co-workers in 1963 and the surface concentration of the species was measured to be 1.22 into 10 to the power of minus 5 moles per centimeter cube. So now let us calculate the y-spreader parameter from this expression and then see whether the internal diffusional limitations exist or not. So if I calculate the y-spreader parameter so Cwp which is basically the minus rA observed that is the observed reaction rate multiplied by the density of the catalyst into r square divided by the density effective diffusivity of the species multiplied by CAS that turns out to be about 1.88 into 10 power minus 3 which is significantly smaller than 1. Now this would mean that this particular reaction does not have a diffusional limitation. So this suggests that, suggests no diffusional limitation that means that the overall reaction is not controlled by the internal diffusional limitations. So which means that it suggests that there is no internal diffusional limitations for this particular reaction carbon and the carbon dioxide heterogeneous reaction it leads to two molecules of carbon monoxide. However, after the reaction was conducted the same researchers they actually cut open the catalyst and measured, looked at the carbon consumption profile, measured the carbon consumption profile in the catalyst and once it was measured it suggested the profile actually suggested that there was strong internal diffusional limitations. It suggested that the reaction was strongly limited by the internal diffusion and that shows that the y-spreader criterion does not work for this reaction. This reaction where the carbon reacts with carbon dioxide to form two molecules of carbon monoxide in this particular case the y-spreader criterion does not work, it does not predict correctly the presence of the internal diffusional limitations. So the question is what is the, so because it does not predict one needs to find out what is the corrective measure for this and what is the correct criteria, what is the correct generalized criteria in order to estimate from the experimental observation whether the external internal diffusional limitations are present or not for a given heterogeneous catalytic reaction. So clearly there is a need for a different framework, why do we need a different framework for this reaction? Because when we look at the mechanism of this particular reaction we look when you go deep into the mechanism and we try to understand the mechanism of this particular reaction it was observed that the carbon monoxide which is a product it strongly absorbs on to the catalyst side and then it inhibits the reaction. So clearly this mechanism suggests that the mechanism behind this heterogeneous catalytic reaction it suggests that the carbon monoxide absorbs on to the catalyst sides. So it absorbs on to the catalyst sides and therefore the reaction is strongly inhibited which means that it is not going to follow the classical n-th order reaction, the catalytic reaction is not the rate law is not an n-th order reaction, does not have an n-th order dependence on the concentration of the species because the product is now absorbing on to the catalyst sides and it is strongly inhibiting the reaction. So therefore there is a limitation that is present here. So now the question is if I look at the, now if I look at the Thiele modulus effectiveness factor graph in general and it goes from 0.1 to 10 and this is 1 here and for an n-th order reaction 1st order, 2nd order etc., 0th order, 1st order, 2nd order the eta versus 5 graph it looks like this. On the other hand for other types of rate loss for example adsorption rate law, Langmuir Hinshelwood or L.E.D. Dill type of mechanisms and for exothermic reactions the effectiveness Thiele modulus graph can actually look like this. So therefore when eta is equal to 1, when the Thiele modulus is very small it does not necessarily mean that the effectiveness factor is actually almost equal to 1. So as a result the Weisblatter criterion which hinges on the fact that when for a certain type of reactions the effectiveness factor is almost equal to 1, the Thiele modulus is less than 1. So that factor does not work for the situations where the rate law depends upon or rate law mechanism or rate law follows the Langmuir Hinshelwood type or the L.E.D. Dill type that is when there is an adsorption, product adsorption or adsorption of the species. So heterogeneous reaction, several heterogeneous reactions they actually follow the Langmuir Hinshelwood type kinetics or L.E.D. Dill type kinetics which basically uses the adsorption isotherm which is adsorption isotherm is now incorporated into the rate law. So therefore in these cases the WP criterion the Weisblatter criterion does not work. So now we need to find out what is the generalized criteria which works for all types of rate laws. So the exercise is now to find out what is this generalized criterion. So suppose we define capital phi as eta into phi square and this is now very similar to that of the Weisblatter criterion. This is very similar to CWP that is the Weisblatter parameter. Now in order to come up with a generalized criterion we need to define generalized, we need to find out what is the generalized effectiveness factor, what is the general definition of the effectiveness factor and what is the generalized definition of the Thiele modulus. So the generalized definition for the internal effectiveness factor is basically given by Ra' observed that is the observed reaction rate divided by the corresponding reaction rate on the, at the surface of the catalyst. On the other hand the generalized Thiele modulus can actually be defined as the length scale whatever is the radius of the pellet, effective radius of the pellet into minus rA, that is the reaction rate at the surface concentration multiplied by the density of the catalyst divided by square root of 2 into integral the equilibrium concentration of the species at the center of the catalyst. If the catalyst was of infinite size integral from CA to CAS where CAS is the concentration of the surface into the effective diffusivity of the species DA into minus rA into rho C into dCA that to the power of minus 1 by 2. So that is the generalized Thiele modulus and it should be noted here that CA is the concentration of species at r equal to 0 that is at the center of the catalyst pellet. If the pellet is of infinite size, if the pellet is of infinite size now this quantity CA equilibrium is actually going to be 0 if it is a non-reversible reaction that is it is a forward or a back one of one side reaction then the CA equilibrium here would actually take a value of 0. So if it is a non-reversible reaction then because it is an infinite size pellet then the amount of time that it takes for the species to actually diffuse into the pellet and go all the way to the center will be infinite time and therefore the concentration of the species at the center of the pellet can be assumed to be 0 if it is a non-reversible reaction. Now if it is a reversible reaction then it will be an equilibrium concentration. So now if I look at this expression if I plug in the generalized effectiveness factor in the generalized Thiele modulus expression into this modified or new generalized criterion may be eta phi square and that should be equal to minus rA that is the observed rate divided by at the surface at the rate at the surface concentration multiplied by r square into minus rA s square into rho c square divided by 2 times integral CA equilibrium that is the equilibrium concentration at the center into CA s into the effective diffusivity of that species into minus rA into rho c into dc into dca. So that is the expression for the modified or generalized criterion for deciding whether the internal diffusion is going to exist or not. Internal diffusion limitation is going to exist or not. So suppose if this quantity so now we can rewrite this as by cancelling some of these terms. So you can cancel this term with this square and this case assume that the density of the catalyst does not change then we can cancel off these and so we can rewrite this as the observed rate into r square that is the square of the length scale that is the rate evaluated at the surface concentration multiplied by the density divided by 2 times integral CA equilibrium that is the equilibrium concentration of the species at the center of the pellet if the pellet is infinitely large and the integral goes from equilibrium concentration to the at r equal to 0 to the surface concentration multiplied by the corresponding diffusivity into the reaction rate into dca. So now the modified criterion is that if this quantity phi is less than 1 then there is no internal diffusional limitations. In fact this quantity here this generalized modulus here if we plug in the rate expression for an nth order reaction it actually reduces to the Weisz's prater criterion. So therefore this is a generalized model which also includes the Weisz's prater criterion of Weisz's prater criterion which is used for deciding whether the internal diffusional limitation is present or absent based on the experimental data. Now let us look at the specific example we had initiated today that is the reaction of C plus CO2 giving 2 times CO. So let us look at what happens what is the actual whether the diffusional limitation is actually predicted by the modified or generalized criterion. So remember that the Weisz's prater criterion did not predict the presence of the internal diffusional limitations. However by cutting open the catalyst the experimental evidence by looking at the profile of the carbon content such as that the diffusional limitations was strongly present and it strongly inhibited the reaction. So let us now plug in the rate law into the generalized modulus and look at whether the internal diffusional limitations was present or absent. So the reaction scheme is B plus A giving 2 times D so species B is the carbon and A is CO2 and D is the product carbon monoxide. So the reaction rate law by looking at the detailed mechanisms that is involved in the heterogeneous catalytic reaction has been found to be K into CA divided by 1 plus K2 into CD that is the adsorption constant for adsorption equilibrium constant for carbon monoxide plus K3 into CA. This is the adsorption constant correspondingly for the carbon dioxide and so if we assume that the diffusivity, effective diffusivity of species A is equal to the effective diffusivity of species D and by assuming that it is a equimolar counter diffusion system and if we also assume that the concentration of the product species which is carbon monoxide at the surface is approximately 0. Now this is valid because the experiment suggests that there is a strong adsorption of the product onto the catalyst. So therefore we can expect that the amount of carbon monoxide which is actually left the catalyst and amount that is present on the surface is negligible. So therefore we can assume that the concentration of DS on the surface is approximately equal to 0. So therefore using these assumptions the rate law can now be rewritten as K into CA divided by 1 plus 2 K2 CAS that is the concentration of carbon dioxide at the surface of the catalyst plus K3 which is the equilibrium constant for the adsorption of CO2 minus 2 times K2 that is the equilibrium constant for adsorption of carbon monoxide, product carbon monoxide onto the surface of the catalyst multiplied by the concentration of the species CA. So now integrating the expression for the generalized modulus we will find that phi capital phi which is generalized modulus generalized criterion that is equal to internal effectiveness factor multiplied by phi square. Now because it is not a reversible reaction we can assume that CA equilibrium is equal to 0. So if we assume that CA equilibrium that is the concentration of the species that is carbon dioxide at the centre of the pellet if the pellet was infinitely long if that is approximately equal to 0 because it is a non-reversible reaction. And therefore phi is equal to minus rA prime the observed reaction rate multiplied by r square into rho C evaluated at the surface divided by 2 times integral 0 to CAS that is the integral into the diffusivity of species DEA into minus rA prime into DC. So now we can plug in the rate expression here remember that the rate expression is given by the K times CA divided by 1 plus 2 times K3 into K2 into CA plus K3 minus 2 K2 into CA. So the first one is the product of 1 plus 2 K3 so the rA is given by minus rA prime is given by K into CA divided by 1 plus 2 K3 to 1 plus 2 K2 into CAS plus K3 minus 2 K2 into CA. So that is the reaction rate. So now if we can that is the rate law for the catalytic reaction now plug in this rate law so we can plug in this rate law into this expression here and then we can integrate the expression. So performing the integration it turns out that phi is equal to observed into the density of the catalyst multiplied by r square divided by 2 times the corresponding effective diffusivity DEA into 1 plus K3 CAS divided by K3 minus 2 K2 into 1 minus 1 plus 2 K2 CAS divided by CA 3 into K3 minus 2 K2 multiplied by logarithm of 1 plus K3 CAS divided by 1 plus 2 K2 CAS inverse of this. So that is the expression for the modified generalized criterion phi it is equal to Eta internal effectiveness factor multiplied by the corresponding Thiele modulus multiplied by phi square. So now the same experimental group Austin Walker's group they have also while performing these experiments they estimated that K2 which is the adsorption constant for carbon monoxide is given by 4.15 into 10 to the power of 9 centimeter cube per mole. Similarly K3 was also estimated as 3.38 into 10 to the power of 5 centimeter cube per mole. So plugging in these numbers we can find that the generalized phi which is equal to Eta times phi square which is the parameter in the generalized criterion that should be equal to 2.5 which is certainly greater than 1. So clearly the generalized model generalized criterion suggest that there is strong internal strong internal diffusional limitations strong internal diffusion limitations. In fact that is what was observed experimentally. So that was what was the experimental observation as well. So therefore in order to find out whether there is internal diffusion limitation or not depending upon what is the nature of the rate law a simple vice predator criterion can be used if it is a simple nth order reaction. But if it is the rate law is not as simple as that then one has to use these generalized criterion phi which is given by earlier expression that we just derived where one needs to find out what is integrate the expression of the diffusivity multiplied by the rate going from the concentration of the species at the center of the pellet if it is infinitely long all the way up to the concentration of the species at the surface of the catalyst. So now let us look at what if there is network of first order reactions. So this is we looked at what is the experimental criteria and what is the Thiele modulus and effectiveness factor and what is their relationship and how to use that information in order to find out whether there is internal diffusion limitations or not for a single reaction. Suppose if there is network of first order reactions then can we develop a is there a general framework in order to find out what is the effectiveness factor and what is the Thiele modulus for each of these first order reactions and it is very useful in in in practice because the diffusional limitations of one species can now strongly affect the affect the selectivity of the of the of the of the desired product. So therefore it is important to understand what is the Thiele modulus of each of these species for each of the species and the corresponding effectiveness factor. So let us now look at what is the general framework. So this this was actually done by Bischoff in 1967. So now suppose if the first set of network of first order reactions was carried out in a porous catalyst where all the species which is which are reactants they are they diffuse into the catalyst and moment they diffuse into the catalyst the reaction happens at some of these species they can adsorpt on to the surface of the catalyst sites and then the reaction happens on the catalyst site and when the reaction is completed the product actually desorbs if it is in the if it is still adsorpt on to the active sites then it desorbs on from the surface and the product leaves the catalyst. Now it may be that some of these species directly go into the gas stream and so they leave the catalyst without the desorption step which may be present. So therefore suppose if I assume that ai are the n species which is participating in this network of first order reactions. So ai for all i going from 1 to n. So there are n species which are present and n species which is participating in this network of first order reactions. Now if Cj for all j going from 1 to n is the concentration of this species concentration of species j Cj is the concentration of species j for all values going from j j equal to 1 to n. So now the local rate the local rate for species ai because it is a network of first order reactions. So the local reaction rate for species ai is given by ri which is the rate ri that will be some j equal to 1 to n and j not equal to i kij into Cj minus kji into Ci. Now kij so this term here corresponds to the rate of reaction where species j is converted to species i. So basically here the reactant is species j and the product is species i. Now the second term here this corresponds to the rate of reaction where species i leads to formation of species j. So that is the reaction rate. So this is the rate of reaction where species i leads to formation of species j and the first term corresponds to the rate of reaction where the species j is consumed and species i is the product that is formed. So now if I assume that all kij they are all first order rate constants and they have units of time inverse and it is important to note that there cannot be a situation where the species j is converted to itself that is why this summation does not include the summation should not include the ith species. So therefore it is represented as j not equal to i that is this summation is for n minus 1 species with the j equal to i is not included in this summation. So now if we write a if we assume that the diffusivity of each of these species is di. So if di is the diffusivity of ai, di is the diffusivity of species ai then one can write a mole balance for species ai. So one can write a mole balance that incorporates diffusion and reaction mole balance for ai incorporating diffusion and reaction that incorporates diffusion and reaction. So the mole balance will be minus di into del square i. So this is the Laplacian in the in particular coordinate system whichever coordinates in which the reaction is whichever coordinates the pellet is actually designed or the geometry of the pellet and that should be equal to 1 to n j not equal to i kij Cj minus kji Ci. Now del square suppose if it is spherical coordinates if it is a spherical catalyst then del square will be 1 by r square into d by dr into r square d by dr. So that is the that is the Laplacian in spherical coordinates if the catalyst were to be a spherical particle and remember that the first term here corresponds to the species reactant being j and the product which is formed is species i and the second term corresponds to the reactant being species i which is being consumed in order to form a product j. So that is the nomenclature that will be used for demonstrating the Thiele modulus and effectiveness factor for network of first order reactions. Now this mole balance is valid for each and every species i for all n species and so one can write this in the vectorial form. In the vector form this can be written as the diffusivity D multiplied by the Laplacian diffusivity matrix D multiplied by Laplacian of the concentration vector C that should be equal to the rate constant matrix K multiplied by the concentration vector C. So now the diffusivity matrix is essentially a diagonal matrix it is a diagonal matrix and that looks like D1, Dn, D2. So it is an n cross n matrix where the diagonal elements are the diffusivity of each of the molecular n molecular species. Now similarly the concentration C the concentration vector can be written as the concentration vector C is essentially a vector of concentration C1, C2 etc up to Cn. So that is a n cross 1 vector so n rows and 1 column. So n cross 1 vector we are containing the concentration of this n species which is actually participating in the network of first order reactions. Then one can look at the rate constant matrix. So the rate constant matrix is essentially looks like this. So where K is rate constant matrix and that is given by sum j equal to 1 to n kj1 where j is not equal to 1 minus k12 all the way up to minus k1n and the second term will be k21 this will be sum j equal to 1 to n kj2 where j is not equal to 2 minus k2n and similarly we can fill this matrix and that will be minus kn1 minus kn2 and that will be sum j equal to 1 to n j is not equal to n kjn. So that is the rate constant matrix. So this contains all the information about the rate constants for first order rate constants for all the reactions which is involved in the network that is being considered. So now one can actually find out so because of the presence of diffusion because of the diffusional limitations the observed kinetics based on the observed reaction rate can be different from what is the actual true kinetics. So the kinetics is falsified because of the presence of the diffusion limitations and that can actually be expressed in terms of vectorial form for the network of first order reactions and that is given by the observed reaction rate constant matrix. Observed rate constant matrix is given by suppose if k observed is the observed rate constant matrix so that should be equal to the true rate constant matrix which is what we just wrote in the last slide and multiplied by the corresponding effectiveness factor matrix. So this is the internal effectiveness factor matrix in the presence of the internal diffusion the reaction rate that is observed is actually falsified and the observed reaction rate which is given by this matrix which contains an n cross n matrix containing all the reaction kinetics rate constants and that is given by the true rate constants multiplied by the corresponding effectiveness factor matrix. So what is this effectiveness factor vector? So it can be defined as the if we solve the equations and find out what is the effectiveness factor. So the effectiveness factor matrix is given by 3 that is the matrix of Thiele modulus it is the square of the inverse inverse of the square of the matrix of Thiele modulus multiplied by matrix of Thiele moduli into cot hyperbolic of the itself matrix of cot hyperbolic minus the identity matrix. So this is the Thiele modulus so phi is the Thiele modulus matrix and this is the cot hyperbolic of the Thiele modulus matrix so that is the that is the matrix of cot hyperbolic and it is a diagonal matrix and the i is the identity matrix and eta is the effectiveness factor matrix, theta is the internal effectiveness factor matrix and this is essentially a diagonal matrix and this is diagonal because the Thiele modulus matrix turns out to be a diagonal matrix and the cot hyperbolic function of the diagonal matrix also is a diagonal matrix and therefore the effectiveness factor matrix is also a diagonal matrix consisting of the individual effectiveness factor of each of these network of reactions. So now the cot hyperbolic of the Thiele modulus matrix that is a matrix is given by it is a diagonal matrix it is the cot hyperbolic of phi 1, 0, cot hyperbolic of phi 2, etc., cot hyperbolic of phi n. So that is a diagonal consisting of the cot hyperbolic of each of the Thiele modulus corresponding to each of these species and the overall Thiele modulus matrix which is again a diagonal matrix square of that is given by r square which is the length scale of the catalyst multiplied by the diffusivity matrix which is again a diagonal matrix inverse of that multiplied by the corresponding rate constant matrix first on network of first order reaction rate constant matrix. So this is again a diagonal matrix this is again a diagonal matrix and so the network of this once we know the Thiele modulus matrix we should be able to find out what is the cot hyperbolic and we can substitute that in the expression for the relationship between the Thiele modulus matrix and the effectiveness factor matrix and from that the effectiveness factor matrix can be found out and using that one can actually find out what is the actual observed kinetics and express that in terms of the true kinetics. So from experiments if we measure the actual kinetics and from the effectiveness factor we will be able to use that expression to find out what is the true kinetics of the network of first order reactions. So this is important because the diffusional effects strongly affect the selectivity of the product that is designed. So the diffusional effects they affect selectivity and so because the effectiveness factor matrix is a diagonal and the Thiele modulus matrix is also a diagonal matrix one can easily deduce that the species with that has smallest eta smallest internal effectiveness factor will actually have the largest Thiele modulus. So species n whose Thiele modulus is the largest will have the correspondingly smallest effectiveness factor that is that can be deduce simply from the expressions. So now let us look at what are all the experimental limiting cases, what how to deduce these limiting cases from the experimental data. So if you want to summarize what are the features of the experiments or what are the information from the experiments that needs to be used in order to deduce whether a particular limiting case exists in a given heterogeneous catalytic reaction. So that can be summarized quite nicely depending upon the dependence of the rate on various parameters or various system parameters. So let us look at the limiting cases from experimental data look at the limiting cases from experimental data. So suppose if you look at the external mass transport limitations suppose if you look at the external mass transport limitations then the reaction suppose if the reaction is controlled by the external mass transfer then the reaction rate minus rA is given by the mass transport coefficient Kc multiplied by the area per unit volume of the catalyst into the concentration of the species the bulk concentration of the species. So where Kc is the mass transport coefficient and this can typically be estimated using various correlations appropriate correlations for example one could use a Thornus-Kramer correlation. So one could use a Thornus-Kramer correlation in order to estimate what is the mass transport coefficient and the Ac is the area per unit volume of the catalyst and Ca is the concentration bulk concentration of the species. Now if we look at these the dependence of the mass transport coefficient on various system parameters so we could now look at the Thornus-Kramer correlation because the mass transport is given by these correlations. So let us take an example and look at the Thornus-Kramer relationship so that will be that is given by the Sherwood number is equal to the Reynolds number based on the particle diameter so to the power of half multiplied by the Schmidt number to the power of 1 by 3. So that is the dependence of the Sherwood number on the Reynolds number of the part based on the particle diameter multiplied by the Schmidt number. What is Reynolds number? Reynolds number is given by the superficial velocity u multiplied by the diameter of the particle dp divided by 1 minus porosity into the kinematic viscosity. Remember that phi here is not Thiele modulus this is the porosity of the catalyst bed in which the reaction is being conducted and the Schmidt number Sc is given by kinematic viscosity divided by the diffusivity of that species, molecular diffusivity of that species and so from here and Sherwood number is given by mass transport coefficient Kc multiplied by the diameter of the particle dp divided by the corresponding diffusivity into phi by 1 minus phi. Once again here phi refers to the porosity of the bed, phi refers to the porosity of the bed. So from here we can substitute these expressions into the Thornus-Kramer relationship so this is the Thornus-Kramer relationship. So from here we can see that Kc dp by dAb into phi by 1 minus phi that should be equal to u into dp divided by 1 minus phi into nu to the power of half into nu by d to the power of 1 by 3. So now from here we can deduce that the mass transport coefficient Kc is a function of square root of dp which appears in the Reynolds number term and then if we bring this dp, if we divide this expression by dp so we will find that we can bring this to the denominator and we will find that the mass transport coefficient Kc is now a function of Kc is now proportional to 1 by square root of dp. Now in addition to this the mass transport coefficient is proportional to square root of the superficial velocity. Now the surface area per unit volume of the catalyst is essentially proportional to 1 by dp because it is the surface area per unit volume and therefore we can say that the reaction rate of that particular species is proportional to 1 by square root of dp into 1 by diameter of the particle and that is equal to 1 by dp to the power of 3 by 2. And the mass transport coefficient Kc is proportional to the temperature at which the reaction is being conducted so which means that the reaction rate is now proportional to temperature. So what we have looked at in this lecture so far is we have looked at the generalized criterion for determining what is the whether based on the experimental data whether the internal diffusion controls the overall catalytic reaction heterogeneous catalytic reaction. And then we had looked at what if there is a network of first order reactions what is the general framework for finding the effectiveness factor and theory modulus of various species that participate in a network of first order reaction. And then we initiated discussion on how to use experimental data and to identify what are the various kinds of limitations and how the rate depends upon various systems parameter and so we will continue with this in the next lecture. Thank you.